Math 220 Assignment

November 5, 2001

Due Wednesday, November 7

Let U denote the 3 \times 3 matrix

 {1}/{3} 
(
2 
2
1
-2
1
2
-1
2
-2
)
  , 

and let phi be the linear function from R^{3} to R^{3} defined by phi(x) = U x for all x in R^{3}.

  1. Show that the columns of U are mutually perpendicular vectors in R^{3} of length 1.

  2. Show that the rows of the transposed matrix U^{t} are mutually perpendicular vectors in R^{3} of length 1.

  3. Compute the matrix product U^{t} U.

  4. Show that phi is an invertible linear function, and find the matrix for phi^{-1}.

  5. Explain why the function phi preserves lengths and angles. Hint. What effect does applying phi have on the ``dot'' product of two vectors?

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